src/HOL/Algebra/UnivPoly.thy
author berghofe
Fri Jul 01 14:03:50 2005 +0200 (2005-07-01)
changeset 16639 5a89d3622ac0
parent 16417 9bc16273c2d4
child 17094 7a3c2efecffe
permissions -rw-r--r--
Removed setsubgoaler hack (thanks to strengthened finsum_cong).
ballarin@13940
     1
(*
wenzelm@14706
     2
  Title:     HOL/Algebra/UnivPoly.thy
ballarin@13940
     3
  Id:        $Id$
ballarin@13940
     4
  Author:    Clemens Ballarin, started 9 December 1996
ballarin@13940
     5
  Copyright: Clemens Ballarin
ballarin@13940
     6
*)
ballarin@13940
     7
wenzelm@14577
     8
header {* Univariate Polynomials *}
ballarin@13940
     9
haftmann@16417
    10
theory UnivPoly imports Module begin
ballarin@13940
    11
ballarin@14553
    12
text {*
wenzelm@14666
    13
  Polynomials are formalised as modules with additional operations for
wenzelm@14666
    14
  extracting coefficients from polynomials and for obtaining monomials
wenzelm@14666
    15
  from coefficients and exponents (record @{text "up_ring"}).  The
wenzelm@14666
    16
  carrier set is a set of bounded functions from Nat to the
wenzelm@14666
    17
  coefficient domain.  Bounded means that these functions return zero
wenzelm@14666
    18
  above a certain bound (the degree).  There is a chapter on the
wenzelm@14706
    19
  formalisation of polynomials in the PhD thesis \cite{Ballarin:1999},
wenzelm@14706
    20
  which was implemented with axiomatic type classes.  This was later
wenzelm@14706
    21
  ported to Locales.
ballarin@14553
    22
*}
ballarin@14553
    23
wenzelm@14666
    24
ballarin@13949
    25
subsection {* The Constructor for Univariate Polynomials *}
ballarin@13940
    26
ballarin@15095
    27
text {*
ballarin@15095
    28
  Functions with finite support.
ballarin@15095
    29
*}
ballarin@15095
    30
wenzelm@14666
    31
locale bound =
wenzelm@14666
    32
  fixes z :: 'a
wenzelm@14666
    33
    and n :: nat
wenzelm@14666
    34
    and f :: "nat => 'a"
wenzelm@14666
    35
  assumes bound: "!!m. n < m \<Longrightarrow> f m = z"
ballarin@13940
    36
wenzelm@14666
    37
declare bound.intro [intro!]
wenzelm@14666
    38
  and bound.bound [dest]
ballarin@13940
    39
ballarin@13940
    40
lemma bound_below:
wenzelm@14666
    41
  assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"
ballarin@13940
    42
proof (rule classical)
ballarin@13940
    43
  assume "~ ?thesis"
ballarin@13940
    44
  then have "m < n" by arith
ballarin@13940
    45
  with bound have "f n = z" ..
ballarin@13940
    46
  with nonzero show ?thesis by contradiction
ballarin@13940
    47
qed
ballarin@13940
    48
ballarin@13940
    49
record ('a, 'p) up_ring = "('a, 'p) module" +
ballarin@13940
    50
  monom :: "['a, nat] => 'p"
ballarin@13940
    51
  coeff :: "['p, nat] => 'a"
ballarin@13940
    52
wenzelm@14651
    53
constdefs (structure R)
ballarin@15095
    54
  up :: "('a, 'm) ring_scheme => (nat => 'a) set"
wenzelm@14651
    55
  "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}"
ballarin@15095
    56
  UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
ballarin@13940
    57
  "UP R == (|
ballarin@13940
    58
    carrier = up R,
wenzelm@14651
    59
    mult = (%p:up R. %q:up R. %n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)),
wenzelm@14651
    60
    one = (%i. if i=0 then \<one> else \<zero>),
wenzelm@14651
    61
    zero = (%i. \<zero>),
wenzelm@14651
    62
    add = (%p:up R. %q:up R. %i. p i \<oplus> q i),
wenzelm@14651
    63
    smult = (%a:carrier R. %p:up R. %i. a \<otimes> p i),
wenzelm@14651
    64
    monom = (%a:carrier R. %n i. if i=n then a else \<zero>),
ballarin@13940
    65
    coeff = (%p:up R. %n. p n) |)"
ballarin@13940
    66
ballarin@13940
    67
text {*
ballarin@13940
    68
  Properties of the set of polynomials @{term up}.
ballarin@13940
    69
*}
ballarin@13940
    70
ballarin@13940
    71
lemma mem_upI [intro]:
ballarin@13940
    72
  "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
ballarin@13940
    73
  by (simp add: up_def Pi_def)
ballarin@13940
    74
ballarin@13940
    75
lemma mem_upD [dest]:
ballarin@13940
    76
  "f \<in> up R ==> f n \<in> carrier R"
ballarin@13940
    77
  by (simp add: up_def Pi_def)
ballarin@13940
    78
ballarin@13940
    79
lemma (in cring) bound_upD [dest]:
ballarin@13940
    80
  "f \<in> up R ==> EX n. bound \<zero> n f"
ballarin@13940
    81
  by (simp add: up_def)
ballarin@13940
    82
ballarin@13940
    83
lemma (in cring) up_one_closed:
ballarin@13940
    84
   "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"
ballarin@13940
    85
  using up_def by force
ballarin@13940
    86
ballarin@13940
    87
lemma (in cring) up_smult_closed:
ballarin@13940
    88
  "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"
ballarin@13940
    89
  by force
ballarin@13940
    90
ballarin@13940
    91
lemma (in cring) up_add_closed:
ballarin@13940
    92
  "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"
ballarin@13940
    93
proof
ballarin@13940
    94
  fix n
ballarin@13940
    95
  assume "p \<in> up R" and "q \<in> up R"
ballarin@13940
    96
  then show "p n \<oplus> q n \<in> carrier R"
ballarin@13940
    97
    by auto
ballarin@13940
    98
next
ballarin@13940
    99
  assume UP: "p \<in> up R" "q \<in> up R"
ballarin@13940
   100
  show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"
ballarin@13940
   101
  proof -
ballarin@13940
   102
    from UP obtain n where boundn: "bound \<zero> n p" by fast
ballarin@13940
   103
    from UP obtain m where boundm: "bound \<zero> m q" by fast
ballarin@13940
   104
    have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"
ballarin@13940
   105
    proof
ballarin@13940
   106
      fix i
ballarin@13940
   107
      assume "max n m < i"
ballarin@13940
   108
      with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp
ballarin@13940
   109
    qed
ballarin@13940
   110
    then show ?thesis ..
ballarin@13940
   111
  qed
ballarin@13940
   112
qed
ballarin@13940
   113
ballarin@13940
   114
lemma (in cring) up_a_inv_closed:
ballarin@13940
   115
  "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
ballarin@13940
   116
proof
ballarin@13940
   117
  assume R: "p \<in> up R"
ballarin@13940
   118
  then obtain n where "bound \<zero> n p" by auto
ballarin@13940
   119
  then have "bound \<zero> n (%i. \<ominus> p i)" by auto
ballarin@13940
   120
  then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
ballarin@13940
   121
qed auto
ballarin@13940
   122
ballarin@13940
   123
lemma (in cring) up_mult_closed:
ballarin@13940
   124
  "[| p \<in> up R; q \<in> up R |] ==>
wenzelm@14666
   125
  (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"
ballarin@13940
   126
proof
ballarin@13940
   127
  fix n
ballarin@13940
   128
  assume "p \<in> up R" "q \<in> up R"
wenzelm@14666
   129
  then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"
ballarin@13940
   130
    by (simp add: mem_upD  funcsetI)
ballarin@13940
   131
next
ballarin@13940
   132
  assume UP: "p \<in> up R" "q \<in> up R"
wenzelm@14666
   133
  show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"
ballarin@13940
   134
  proof -
ballarin@13940
   135
    from UP obtain n where boundn: "bound \<zero> n p" by fast
ballarin@13940
   136
    from UP obtain m where boundm: "bound \<zero> m q" by fast
wenzelm@14666
   137
    have "bound \<zero> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
ballarin@13940
   138
    proof
wenzelm@14666
   139
      fix k assume bound: "n + m < k"
ballarin@13940
   140
      {
wenzelm@14666
   141
        fix i
wenzelm@14666
   142
        have "p i \<otimes> q (k-i) = \<zero>"
wenzelm@14666
   143
        proof (cases "n < i")
wenzelm@14666
   144
          case True
wenzelm@14666
   145
          with boundn have "p i = \<zero>" by auto
ballarin@13940
   146
          moreover from UP have "q (k-i) \<in> carrier R" by auto
wenzelm@14666
   147
          ultimately show ?thesis by simp
wenzelm@14666
   148
        next
wenzelm@14666
   149
          case False
wenzelm@14666
   150
          with bound have "m < k-i" by arith
wenzelm@14666
   151
          with boundm have "q (k-i) = \<zero>" by auto
wenzelm@14666
   152
          moreover from UP have "p i \<in> carrier R" by auto
wenzelm@14666
   153
          ultimately show ?thesis by simp
wenzelm@14666
   154
        qed
ballarin@13940
   155
      }
wenzelm@14666
   156
      then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"
wenzelm@14666
   157
        by (simp add: Pi_def)
ballarin@13940
   158
    qed
ballarin@13940
   159
    then show ?thesis by fast
ballarin@13940
   160
  qed
ballarin@13940
   161
qed
ballarin@13940
   162
wenzelm@14666
   163
ballarin@13940
   164
subsection {* Effect of operations on coefficients *}
ballarin@13940
   165
ballarin@13940
   166
locale UP = struct R + struct P +
ballarin@13940
   167
  defines P_def: "P == UP R"
ballarin@13940
   168
ballarin@13940
   169
locale UP_cring = UP + cring R
ballarin@13940
   170
ballarin@13940
   171
locale UP_domain = UP_cring + "domain" R
ballarin@13940
   172
ballarin@13940
   173
text {*
ballarin@15095
   174
  Temporarily declare @{thm [locale=UP] P_def} as simp rule.
ballarin@15095
   175
*}
ballarin@13940
   176
ballarin@13940
   177
declare (in UP) P_def [simp]
ballarin@13940
   178
ballarin@13940
   179
lemma (in UP_cring) coeff_monom [simp]:
ballarin@13940
   180
  "a \<in> carrier R ==>
ballarin@13940
   181
  coeff P (monom P a m) n = (if m=n then a else \<zero>)"
ballarin@13940
   182
proof -
ballarin@13940
   183
  assume R: "a \<in> carrier R"
ballarin@13940
   184
  then have "(%n. if n = m then a else \<zero>) \<in> up R"
ballarin@13940
   185
    using up_def by force
ballarin@13940
   186
  with R show ?thesis by (simp add: UP_def)
ballarin@13940
   187
qed
ballarin@13940
   188
ballarin@13940
   189
lemma (in UP_cring) coeff_zero [simp]:
ballarin@15095
   190
  "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>"
ballarin@13940
   191
  by (auto simp add: UP_def)
ballarin@13940
   192
ballarin@13940
   193
lemma (in UP_cring) coeff_one [simp]:
ballarin@15095
   194
  "coeff P \<one>\<^bsub>P\<^esub> n = (if n=0 then \<one> else \<zero>)"
ballarin@13940
   195
  using up_one_closed by (simp add: UP_def)
ballarin@13940
   196
ballarin@13940
   197
lemma (in UP_cring) coeff_smult [simp]:
ballarin@13940
   198
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   199
  coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"
ballarin@13940
   200
  by (simp add: UP_def up_smult_closed)
ballarin@13940
   201
ballarin@13940
   202
lemma (in UP_cring) coeff_add [simp]:
ballarin@13940
   203
  "[| p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   204
  coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"
ballarin@13940
   205
  by (simp add: UP_def up_add_closed)
ballarin@13940
   206
ballarin@13940
   207
lemma (in UP_cring) coeff_mult [simp]:
ballarin@13940
   208
  "[| p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   209
  coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"
ballarin@13940
   210
  by (simp add: UP_def up_mult_closed)
ballarin@13940
   211
ballarin@13940
   212
lemma (in UP) up_eqI:
ballarin@13940
   213
  assumes prem: "!!n. coeff P p n = coeff P q n"
ballarin@13940
   214
    and R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@13940
   215
  shows "p = q"
ballarin@13940
   216
proof
ballarin@13940
   217
  fix x
ballarin@13940
   218
  from prem and R show "p x = q x" by (simp add: UP_def)
ballarin@13940
   219
qed
wenzelm@14666
   220
ballarin@13940
   221
subsection {* Polynomials form a commutative ring. *}
ballarin@13940
   222
wenzelm@14666
   223
text {* Operations are closed over @{term P}. *}
ballarin@13940
   224
ballarin@13940
   225
lemma (in UP_cring) UP_mult_closed [simp]:
ballarin@15095
   226
  "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P"
ballarin@13940
   227
  by (simp add: UP_def up_mult_closed)
ballarin@13940
   228
ballarin@13940
   229
lemma (in UP_cring) UP_one_closed [simp]:
ballarin@15095
   230
  "\<one>\<^bsub>P\<^esub> \<in> carrier P"
ballarin@13940
   231
  by (simp add: UP_def up_one_closed)
ballarin@13940
   232
ballarin@13940
   233
lemma (in UP_cring) UP_zero_closed [intro, simp]:
ballarin@15095
   234
  "\<zero>\<^bsub>P\<^esub> \<in> carrier P"
ballarin@13940
   235
  by (auto simp add: UP_def)
ballarin@13940
   236
ballarin@13940
   237
lemma (in UP_cring) UP_a_closed [intro, simp]:
ballarin@15095
   238
  "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P"
ballarin@13940
   239
  by (simp add: UP_def up_add_closed)
ballarin@13940
   240
ballarin@13940
   241
lemma (in UP_cring) monom_closed [simp]:
ballarin@13940
   242
  "a \<in> carrier R ==> monom P a n \<in> carrier P"
ballarin@13940
   243
  by (auto simp add: UP_def up_def Pi_def)
ballarin@13940
   244
ballarin@13940
   245
lemma (in UP_cring) UP_smult_closed [simp]:
ballarin@15095
   246
  "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^bsub>P\<^esub> p \<in> carrier P"
ballarin@13940
   247
  by (simp add: UP_def up_smult_closed)
ballarin@13940
   248
ballarin@13940
   249
lemma (in UP) coeff_closed [simp]:
ballarin@13940
   250
  "p \<in> carrier P ==> coeff P p n \<in> carrier R"
ballarin@13940
   251
  by (auto simp add: UP_def)
ballarin@13940
   252
ballarin@13940
   253
declare (in UP) P_def [simp del]
ballarin@13940
   254
ballarin@13940
   255
text {* Algebraic ring properties *}
ballarin@13940
   256
ballarin@13940
   257
lemma (in UP_cring) UP_a_assoc:
ballarin@13940
   258
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@15095
   259
  shows "(p \<oplus>\<^bsub>P\<^esub> q) \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)"
ballarin@13940
   260
  by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
ballarin@13940
   261
ballarin@13940
   262
lemma (in UP_cring) UP_l_zero [simp]:
ballarin@13940
   263
  assumes R: "p \<in> carrier P"
ballarin@15095
   264
  shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p"
ballarin@13940
   265
  by (rule up_eqI, simp_all add: R)
ballarin@13940
   266
ballarin@13940
   267
lemma (in UP_cring) UP_l_neg_ex:
ballarin@13940
   268
  assumes R: "p \<in> carrier P"
ballarin@15095
   269
  shows "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   270
proof -
ballarin@13940
   271
  let ?q = "%i. \<ominus> (p i)"
ballarin@13940
   272
  from R have closed: "?q \<in> carrier P"
ballarin@13940
   273
    by (simp add: UP_def P_def up_a_inv_closed)
ballarin@13940
   274
  from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
ballarin@13940
   275
    by (simp add: UP_def P_def up_a_inv_closed)
ballarin@13940
   276
  show ?thesis
ballarin@13940
   277
  proof
ballarin@15095
   278
    show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   279
      by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
ballarin@13940
   280
  qed (rule closed)
ballarin@13940
   281
qed
ballarin@13940
   282
ballarin@13940
   283
lemma (in UP_cring) UP_a_comm:
ballarin@13940
   284
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   285
  shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p"
ballarin@13940
   286
  by (rule up_eqI, simp add: a_comm R, simp_all add: R)
ballarin@13940
   287
ballarin@13940
   288
lemma (in UP_cring) UP_m_assoc:
ballarin@13940
   289
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@15095
   290
  shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
ballarin@13940
   291
proof (rule up_eqI)
ballarin@13940
   292
  fix n
ballarin@13940
   293
  {
ballarin@13940
   294
    fix k and a b c :: "nat=>'a"
ballarin@13940
   295
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
ballarin@13940
   296
      "c \<in> UNIV -> carrier R"
ballarin@13940
   297
    then have "k <= n ==>
wenzelm@14666
   298
      (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
wenzelm@14666
   299
      (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
wenzelm@14666
   300
      (concl is "?eq k")
ballarin@13940
   301
    proof (induct k)
ballarin@13940
   302
      case 0 then show ?case by (simp add: Pi_def m_assoc)
ballarin@13940
   303
    next
ballarin@13940
   304
      case (Suc k)
ballarin@13940
   305
      then have "k <= n" by arith
ballarin@13940
   306
      then have "?eq k" by (rule Suc)
ballarin@13940
   307
      with R show ?case
wenzelm@14666
   308
        by (simp cong: finsum_cong
ballarin@13940
   309
             add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
ballarin@13940
   310
          (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
ballarin@13940
   311
    qed
ballarin@13940
   312
  }
ballarin@15095
   313
  with R show "coeff P ((p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r) n = coeff P (p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)) n"
ballarin@13940
   314
    by (simp add: Pi_def)
ballarin@13940
   315
qed (simp_all add: R)
ballarin@13940
   316
ballarin@13940
   317
lemma (in UP_cring) UP_l_one [simp]:
ballarin@13940
   318
  assumes R: "p \<in> carrier P"
ballarin@15095
   319
  shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
ballarin@13940
   320
proof (rule up_eqI)
ballarin@13940
   321
  fix n
ballarin@15095
   322
  show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
ballarin@13940
   323
  proof (cases n)
ballarin@13940
   324
    case 0 with R show ?thesis by simp
ballarin@13940
   325
  next
ballarin@13940
   326
    case Suc with R show ?thesis
ballarin@13940
   327
      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
ballarin@13940
   328
  qed
ballarin@13940
   329
qed (simp_all add: R)
ballarin@13940
   330
ballarin@13940
   331
lemma (in UP_cring) UP_l_distr:
ballarin@13940
   332
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@15095
   333
  shows "(p \<oplus>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = (p \<otimes>\<^bsub>P\<^esub> r) \<oplus>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
ballarin@13940
   334
  by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
ballarin@13940
   335
ballarin@13940
   336
lemma (in UP_cring) UP_m_comm:
ballarin@13940
   337
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   338
  shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
ballarin@13940
   339
proof (rule up_eqI)
wenzelm@14666
   340
  fix n
ballarin@13940
   341
  {
ballarin@13940
   342
    fix k and a b :: "nat=>'a"
ballarin@13940
   343
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
wenzelm@14666
   344
    then have "k <= n ==>
wenzelm@14666
   345
      (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
wenzelm@14666
   346
      (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
wenzelm@14666
   347
      (concl is "?eq k")
ballarin@13940
   348
    proof (induct k)
ballarin@13940
   349
      case 0 then show ?case by (simp add: Pi_def)
ballarin@13940
   350
    next
ballarin@13940
   351
      case (Suc k) then show ?case
paulson@15944
   352
        by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
ballarin@13940
   353
    qed
ballarin@13940
   354
  }
ballarin@13940
   355
  note l = this
ballarin@15095
   356
  from R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
ballarin@13940
   357
    apply (simp add: Pi_def)
ballarin@13940
   358
    apply (subst l)
ballarin@13940
   359
    apply (auto simp add: Pi_def)
ballarin@13940
   360
    apply (simp add: m_comm)
ballarin@13940
   361
    done
ballarin@13940
   362
qed (simp_all add: R)
ballarin@13940
   363
ballarin@15596
   364
(*
ballarin@15596
   365
Strange phenomenon in Isar:
ballarin@15596
   366
ballarin@15596
   367
theorem (in UP_cring) UP_cring:
ballarin@15596
   368
  "cring P"
ballarin@15596
   369
proof (rule cringI)
ballarin@15596
   370
  show "abelian_group P" proof (rule abelian_groupI)
ballarin@15596
   371
  fix x y z
ballarin@15596
   372
  assume "x \<in> carrier P" and "y \<in> carrier P" and "z \<in> carrier P"
ballarin@15596
   373
  {
ballarin@15596
   374
  show "x \<oplus>\<^bsub>P\<^esub> y \<in> carrier P" sorry
ballarin@15596
   375
  next
ballarin@15596
   376
  show "x \<oplus>\<^bsub>P\<^esub> y \<oplus>\<^bsub>P\<^esub> z = x \<oplus>\<^bsub>P\<^esub> (y \<oplus>\<^bsub>P\<^esub> z)" sorry
ballarin@15596
   377
  next
ballarin@15596
   378
  show "x \<oplus>\<^bsub>P\<^esub> y = y \<oplus>\<^bsub>P\<^esub> x" sorry
ballarin@15596
   379
  next
ballarin@15596
   380
  show "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> x = x" sorry
ballarin@15596
   381
  next
ballarin@15596
   382
  show "\<exists>y\<in>carrier P. y \<oplus>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>" sorry
ballarin@15596
   383
  next
ballarin@15596
   384
  show "\<zero>\<^bsub>P\<^esub> \<in> carrier P" sorry  last goal rejected!!!
ballarin@15596
   385
*)
ballarin@15596
   386
ballarin@13940
   387
theorem (in UP_cring) UP_cring:
ballarin@13940
   388
  "cring P"
ballarin@13940
   389
  by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
ballarin@13940
   390
    UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
ballarin@13940
   391
ballarin@14399
   392
lemma (in UP_cring) UP_ring:  (* preliminary *)
ballarin@14399
   393
  "ring P"
ballarin@14399
   394
  by (auto intro: ring.intro cring.axioms UP_cring)
ballarin@14399
   395
ballarin@13940
   396
lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
ballarin@15095
   397
  "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
ballarin@13940
   398
  by (rule abelian_group.a_inv_closed
ballarin@14399
   399
    [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   400
ballarin@13940
   401
lemma (in UP_cring) coeff_a_inv [simp]:
ballarin@13940
   402
  assumes R: "p \<in> carrier P"
ballarin@15095
   403
  shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
ballarin@13940
   404
proof -
ballarin@13940
   405
  from R coeff_closed UP_a_inv_closed have
ballarin@15095
   406
    "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^bsub>P\<^esub> p) n)"
ballarin@13940
   407
    by algebra
ballarin@13940
   408
  also from R have "... =  \<ominus> (coeff P p n)"
ballarin@13940
   409
    by (simp del: coeff_add add: coeff_add [THEN sym]
ballarin@14399
   410
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   411
  finally show ?thesis .
ballarin@13940
   412
qed
ballarin@13940
   413
ballarin@13940
   414
text {*
ballarin@13940
   415
  Instantiation of lemmas from @{term cring}.
ballarin@13940
   416
*}
ballarin@13940
   417
ballarin@15095
   418
(* TODO: this should be automated with an instantiation command. *)
ballarin@15095
   419
ballarin@13940
   420
lemma (in UP_cring) UP_monoid:
ballarin@13940
   421
  "monoid P"
ballarin@13940
   422
  by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
ballarin@13940
   423
    UP_cring)
ballarin@13940
   424
(* TODO: provide cring.is_monoid *)
ballarin@13940
   425
ballarin@13940
   426
lemma (in UP_cring) UP_comm_monoid:
ballarin@13940
   427
  "comm_monoid P"
ballarin@13940
   428
  by (fast intro!: cring.is_comm_monoid UP_cring)
ballarin@13940
   429
ballarin@13940
   430
lemma (in UP_cring) UP_abelian_monoid:
ballarin@13940
   431
  "abelian_monoid P"
ballarin@14399
   432
  by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
ballarin@13940
   433
ballarin@13940
   434
lemma (in UP_cring) UP_abelian_group:
ballarin@13940
   435
  "abelian_group P"
ballarin@14399
   436
  by (fast intro!: ring.is_abelian_group UP_ring)
ballarin@13940
   437
ballarin@13940
   438
lemmas (in UP_cring) UP_r_one [simp] =
ballarin@13940
   439
  monoid.r_one [OF UP_monoid]
ballarin@13940
   440
ballarin@13940
   441
lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
ballarin@13940
   442
  monoid.nat_pow_closed [OF UP_monoid]
ballarin@13940
   443
ballarin@13940
   444
lemmas (in UP_cring) UP_nat_pow_0 [simp] =
ballarin@13940
   445
  monoid.nat_pow_0 [OF UP_monoid]
ballarin@13940
   446
ballarin@13940
   447
lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
ballarin@13940
   448
  monoid.nat_pow_Suc [OF UP_monoid]
ballarin@13940
   449
ballarin@13940
   450
lemmas (in UP_cring) UP_nat_pow_one [simp] =
ballarin@13940
   451
  monoid.nat_pow_one [OF UP_monoid]
ballarin@13940
   452
ballarin@13940
   453
lemmas (in UP_cring) UP_nat_pow_mult =
ballarin@13940
   454
  monoid.nat_pow_mult [OF UP_monoid]
ballarin@13940
   455
ballarin@13940
   456
lemmas (in UP_cring) UP_nat_pow_pow =
ballarin@13940
   457
  monoid.nat_pow_pow [OF UP_monoid]
ballarin@13940
   458
ballarin@13940
   459
lemmas (in UP_cring) UP_m_lcomm =
paulson@14963
   460
  comm_monoid.m_lcomm [OF UP_comm_monoid]
ballarin@13940
   461
ballarin@13940
   462
lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
ballarin@13940
   463
ballarin@13940
   464
lemmas (in UP_cring) UP_nat_pow_distr =
ballarin@13940
   465
  comm_monoid.nat_pow_distr [OF UP_comm_monoid]
ballarin@13940
   466
ballarin@13940
   467
lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
ballarin@13940
   468
ballarin@13940
   469
lemmas (in UP_cring) UP_r_zero [simp] =
ballarin@13940
   470
  abelian_monoid.r_zero [OF UP_abelian_monoid]
ballarin@13940
   471
ballarin@13940
   472
lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
ballarin@13940
   473
ballarin@13940
   474
lemmas (in UP_cring) UP_finsum_empty [simp] =
ballarin@13940
   475
  abelian_monoid.finsum_empty [OF UP_abelian_monoid]
ballarin@13940
   476
ballarin@13940
   477
lemmas (in UP_cring) UP_finsum_insert [simp] =
ballarin@13940
   478
  abelian_monoid.finsum_insert [OF UP_abelian_monoid]
ballarin@13940
   479
ballarin@13940
   480
lemmas (in UP_cring) UP_finsum_zero [simp] =
ballarin@13940
   481
  abelian_monoid.finsum_zero [OF UP_abelian_monoid]
ballarin@13940
   482
ballarin@13940
   483
lemmas (in UP_cring) UP_finsum_closed [simp] =
ballarin@13940
   484
  abelian_monoid.finsum_closed [OF UP_abelian_monoid]
ballarin@13940
   485
ballarin@13940
   486
lemmas (in UP_cring) UP_finsum_Un_Int =
ballarin@13940
   487
  abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
ballarin@13940
   488
ballarin@13940
   489
lemmas (in UP_cring) UP_finsum_Un_disjoint =
ballarin@13940
   490
  abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
ballarin@13940
   491
ballarin@13940
   492
lemmas (in UP_cring) UP_finsum_addf =
ballarin@13940
   493
  abelian_monoid.finsum_addf [OF UP_abelian_monoid]
ballarin@13940
   494
ballarin@13940
   495
lemmas (in UP_cring) UP_finsum_cong' =
ballarin@13940
   496
  abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
ballarin@13940
   497
ballarin@13940
   498
lemmas (in UP_cring) UP_finsum_0 [simp] =
ballarin@13940
   499
  abelian_monoid.finsum_0 [OF UP_abelian_monoid]
ballarin@13940
   500
ballarin@13940
   501
lemmas (in UP_cring) UP_finsum_Suc [simp] =
ballarin@13940
   502
  abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
ballarin@13940
   503
ballarin@13940
   504
lemmas (in UP_cring) UP_finsum_Suc2 =
ballarin@13940
   505
  abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
ballarin@13940
   506
ballarin@13940
   507
lemmas (in UP_cring) UP_finsum_add [simp] =
ballarin@13940
   508
  abelian_monoid.finsum_add [OF UP_abelian_monoid]
ballarin@13940
   509
ballarin@13940
   510
lemmas (in UP_cring) UP_finsum_cong =
ballarin@13940
   511
  abelian_monoid.finsum_cong [OF UP_abelian_monoid]
ballarin@13940
   512
ballarin@13940
   513
lemmas (in UP_cring) UP_minus_closed [intro, simp] =
ballarin@13940
   514
  abelian_group.minus_closed [OF UP_abelian_group]
ballarin@13940
   515
ballarin@13940
   516
lemmas (in UP_cring) UP_a_l_cancel [simp] =
ballarin@13940
   517
  abelian_group.a_l_cancel [OF UP_abelian_group]
ballarin@13940
   518
ballarin@13940
   519
lemmas (in UP_cring) UP_a_r_cancel [simp] =
ballarin@13940
   520
  abelian_group.a_r_cancel [OF UP_abelian_group]
ballarin@13940
   521
ballarin@13940
   522
lemmas (in UP_cring) UP_l_neg =
ballarin@13940
   523
  abelian_group.l_neg [OF UP_abelian_group]
ballarin@13940
   524
ballarin@13940
   525
lemmas (in UP_cring) UP_r_neg =
ballarin@13940
   526
  abelian_group.r_neg [OF UP_abelian_group]
ballarin@13940
   527
ballarin@13940
   528
lemmas (in UP_cring) UP_minus_zero [simp] =
ballarin@13940
   529
  abelian_group.minus_zero [OF UP_abelian_group]
ballarin@13940
   530
ballarin@13940
   531
lemmas (in UP_cring) UP_minus_minus [simp] =
ballarin@13940
   532
  abelian_group.minus_minus [OF UP_abelian_group]
ballarin@13940
   533
ballarin@13940
   534
lemmas (in UP_cring) UP_minus_add =
ballarin@13940
   535
  abelian_group.minus_add [OF UP_abelian_group]
ballarin@13940
   536
ballarin@13940
   537
lemmas (in UP_cring) UP_r_neg2 =
ballarin@13940
   538
  abelian_group.r_neg2 [OF UP_abelian_group]
ballarin@13940
   539
ballarin@13940
   540
lemmas (in UP_cring) UP_r_neg1 =
ballarin@13940
   541
  abelian_group.r_neg1 [OF UP_abelian_group]
ballarin@13940
   542
ballarin@13940
   543
lemmas (in UP_cring) UP_r_distr =
ballarin@14399
   544
  ring.r_distr [OF UP_ring]
ballarin@13940
   545
ballarin@13940
   546
lemmas (in UP_cring) UP_l_null [simp] =
ballarin@14399
   547
  ring.l_null [OF UP_ring]
ballarin@13940
   548
ballarin@13940
   549
lemmas (in UP_cring) UP_r_null [simp] =
ballarin@14399
   550
  ring.r_null [OF UP_ring]
ballarin@13940
   551
ballarin@13940
   552
lemmas (in UP_cring) UP_l_minus =
ballarin@14399
   553
  ring.l_minus [OF UP_ring]
ballarin@13940
   554
ballarin@13940
   555
lemmas (in UP_cring) UP_r_minus =
ballarin@14399
   556
  ring.r_minus [OF UP_ring]
ballarin@13940
   557
ballarin@13940
   558
lemmas (in UP_cring) UP_finsum_ldistr =
ballarin@13940
   559
  cring.finsum_ldistr [OF UP_cring]
ballarin@13940
   560
ballarin@13940
   561
lemmas (in UP_cring) UP_finsum_rdistr =
ballarin@13940
   562
  cring.finsum_rdistr [OF UP_cring]
ballarin@13940
   563
wenzelm@14666
   564
ballarin@13940
   565
subsection {* Polynomials form an Algebra *}
ballarin@13940
   566
ballarin@13940
   567
lemma (in UP_cring) UP_smult_l_distr:
ballarin@13940
   568
  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   569
  (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   570
  by (rule up_eqI) (simp_all add: R.l_distr)
ballarin@13940
   571
ballarin@13940
   572
lemma (in UP_cring) UP_smult_r_distr:
ballarin@13940
   573
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   574
  a \<odot>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> a \<odot>\<^bsub>P\<^esub> q"
ballarin@13940
   575
  by (rule up_eqI) (simp_all add: R.r_distr)
ballarin@13940
   576
ballarin@13940
   577
lemma (in UP_cring) UP_smult_assoc1:
ballarin@13940
   578
      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   579
      (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   580
  by (rule up_eqI) (simp_all add: R.m_assoc)
ballarin@13940
   581
ballarin@13940
   582
lemma (in UP_cring) UP_smult_one [simp]:
ballarin@15095
   583
      "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
ballarin@13940
   584
  by (rule up_eqI) simp_all
ballarin@13940
   585
ballarin@13940
   586
lemma (in UP_cring) UP_smult_assoc2:
ballarin@13940
   587
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   588
  (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   589
  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
ballarin@13940
   590
ballarin@13940
   591
text {*
ballarin@13940
   592
  Instantiation of lemmas from @{term algebra}.
ballarin@13940
   593
*}
ballarin@13940
   594
ballarin@15095
   595
(* TODO: this should be automated with an instantiation command. *)
ballarin@15095
   596
ballarin@13940
   597
(* TODO: move to CRing.thy, really a fact missing from the locales package *)
ballarin@13940
   598
lemma (in cring) cring:
ballarin@13940
   599
  "cring R"
ballarin@13940
   600
  by (fast intro: cring.intro prems)
ballarin@13940
   601
ballarin@13940
   602
lemma (in UP_cring) UP_algebra:
ballarin@13940
   603
  "algebra R P"
ballarin@13940
   604
  by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
ballarin@13940
   605
    UP_smult_assoc1 UP_smult_assoc2)
ballarin@13940
   606
ballarin@13940
   607
lemmas (in UP_cring) UP_smult_l_null [simp] =
ballarin@13940
   608
  algebra.smult_l_null [OF UP_algebra]
ballarin@13940
   609
ballarin@13940
   610
lemmas (in UP_cring) UP_smult_r_null [simp] =
ballarin@13940
   611
  algebra.smult_r_null [OF UP_algebra]
ballarin@13940
   612
ballarin@13940
   613
lemmas (in UP_cring) UP_smult_l_minus =
ballarin@13940
   614
  algebra.smult_l_minus [OF UP_algebra]
ballarin@13940
   615
ballarin@13940
   616
lemmas (in UP_cring) UP_smult_r_minus =
ballarin@13940
   617
  algebra.smult_r_minus [OF UP_algebra]
ballarin@13940
   618
ballarin@13949
   619
subsection {* Further lemmas involving monomials *}
ballarin@13940
   620
ballarin@13940
   621
lemma (in UP_cring) monom_zero [simp]:
ballarin@15095
   622
  "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   623
  by (simp add: UP_def P_def)
ballarin@13940
   624
ballarin@13940
   625
lemma (in UP_cring) monom_mult_is_smult:
ballarin@13940
   626
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   627
  shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   628
proof (rule up_eqI)
ballarin@13940
   629
  fix n
ballarin@15095
   630
  have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
ballarin@13940
   631
  proof (cases n)
ballarin@13940
   632
    case 0 with R show ?thesis by (simp add: R.m_comm)
ballarin@13940
   633
  next
ballarin@13940
   634
    case Suc with R show ?thesis
ballarin@13940
   635
      by (simp cong: finsum_cong add: R.r_null Pi_def)
ballarin@13940
   636
        (simp add: m_comm)
ballarin@13940
   637
  qed
ballarin@15095
   638
  with R show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
ballarin@13940
   639
    by (simp add: UP_m_comm)
ballarin@13940
   640
qed (simp_all add: R)
ballarin@13940
   641
ballarin@13940
   642
lemma (in UP_cring) monom_add [simp]:
ballarin@13940
   643
  "[| a \<in> carrier R; b \<in> carrier R |] ==>
ballarin@15095
   644
  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
ballarin@13940
   645
  by (rule up_eqI) simp_all
ballarin@13940
   646
ballarin@13940
   647
lemma (in UP_cring) monom_one_Suc:
ballarin@15095
   648
  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
ballarin@13940
   649
proof (rule up_eqI)
ballarin@13940
   650
  fix k
ballarin@15095
   651
  show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
ballarin@13940
   652
  proof (cases "k = Suc n")
ballarin@13940
   653
    case True show ?thesis
ballarin@13940
   654
    proof -
wenzelm@14666
   655
      from True have less_add_diff:
wenzelm@14666
   656
        "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
ballarin@13940
   657
      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
ballarin@13940
   658
      also from True
nipkow@15045
   659
      have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   660
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   661
        by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
wenzelm@14666
   662
      also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   663
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   664
        by (simp only: ivl_disj_un_singleton)
ballarin@15095
   665
      also from True
ballarin@15095
   666
      have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   667
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   668
        by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
wenzelm@14666
   669
          order_less_imp_not_eq Pi_def)
ballarin@15095
   670
      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
wenzelm@14666
   671
        by (simp add: ivl_disj_un_one)
ballarin@13940
   672
      finally show ?thesis .
ballarin@13940
   673
    qed
ballarin@13940
   674
  next
ballarin@13940
   675
    case False
ballarin@13940
   676
    note neq = False
ballarin@13940
   677
    let ?s =
wenzelm@14666
   678
      "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
ballarin@13940
   679
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
wenzelm@14666
   680
    also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
ballarin@13940
   681
    proof -
ballarin@15095
   682
      have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
ballarin@15095
   683
        by (simp cong: finsum_cong add: Pi_def)
wenzelm@14666
   684
      from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
wenzelm@14666
   685
        by (simp cong: finsum_cong add: Pi_def) arith
nipkow@15045
   686
      have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
wenzelm@14666
   687
        by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
ballarin@13940
   688
      show ?thesis
ballarin@13940
   689
      proof (cases "k < n")
wenzelm@14666
   690
        case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
ballarin@13940
   691
      next
wenzelm@14666
   692
        case False then have n_le_k: "n <= k" by arith
wenzelm@14666
   693
        show ?thesis
wenzelm@14666
   694
        proof (cases "n = k")
wenzelm@14666
   695
          case True
nipkow@15045
   696
          then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
wenzelm@14666
   697
            by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
   698
              ivl_disj_int_singleton Pi_def)
wenzelm@14666
   699
          also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   700
            by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   701
          finally show ?thesis .
wenzelm@14666
   702
        next
wenzelm@14666
   703
          case False with n_le_k have n_less_k: "n < k" by arith
nipkow@15045
   704
          with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
wenzelm@14666
   705
            by (simp add: finsum_Un_disjoint f1 f2
wenzelm@14666
   706
              ivl_disj_int_singleton Pi_def del: Un_insert_right)
wenzelm@14666
   707
          also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
wenzelm@14666
   708
            by (simp only: ivl_disj_un_singleton)
nipkow@15045
   709
          also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
wenzelm@14666
   710
            by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
wenzelm@14666
   711
          also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   712
            by (simp only: ivl_disj_un_one)
wenzelm@14666
   713
          finally show ?thesis .
wenzelm@14666
   714
        qed
ballarin@13940
   715
      qed
ballarin@13940
   716
    qed
ballarin@15095
   717
    also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
ballarin@13940
   718
    finally show ?thesis .
ballarin@13940
   719
  qed
ballarin@13940
   720
qed (simp_all)
ballarin@13940
   721
ballarin@13940
   722
lemma (in UP_cring) monom_mult_smult:
ballarin@15095
   723
  "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
ballarin@13940
   724
  by (rule up_eqI) simp_all
ballarin@13940
   725
ballarin@13940
   726
lemma (in UP_cring) monom_one [simp]:
ballarin@15095
   727
  "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
ballarin@13940
   728
  by (rule up_eqI) simp_all
ballarin@13940
   729
ballarin@13940
   730
lemma (in UP_cring) monom_one_mult:
ballarin@15095
   731
  "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
ballarin@13940
   732
proof (induct n)
ballarin@13940
   733
  case 0 show ?case by simp
ballarin@13940
   734
next
ballarin@13940
   735
  case Suc then show ?case
ballarin@13940
   736
    by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
ballarin@13940
   737
qed
ballarin@13940
   738
ballarin@13940
   739
lemma (in UP_cring) monom_mult [simp]:
ballarin@13940
   740
  assumes R: "a \<in> carrier R" "b \<in> carrier R"
ballarin@15095
   741
  shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
ballarin@13940
   742
proof -
ballarin@13940
   743
  from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
ballarin@15095
   744
  also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"
ballarin@13940
   745
    by (simp add: monom_mult_smult del: r_one)
ballarin@15095
   746
  also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"
ballarin@13940
   747
    by (simp only: monom_one_mult)
ballarin@15095
   748
  also from R have "... = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m))"
ballarin@13940
   749
    by (simp add: UP_smult_assoc1)
ballarin@15095
   750
  also from R have "... = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> (monom P \<one> m \<otimes>\<^bsub>P\<^esub> monom P \<one> n))"
ballarin@13940
   751
    by (simp add: UP_m_comm)
ballarin@15095
   752
  also from R have "... = a \<odot>\<^bsub>P\<^esub> ((b \<odot>\<^bsub>P\<^esub> monom P \<one> m) \<otimes>\<^bsub>P\<^esub> monom P \<one> n)"
ballarin@13940
   753
    by (simp add: UP_smult_assoc2)
ballarin@15095
   754
  also from R have "... = a \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> monom P \<one> m))"
ballarin@13940
   755
    by (simp add: UP_m_comm)
ballarin@15095
   756
  also from R have "... = (a \<odot>\<^bsub>P\<^esub> monom P \<one> n) \<otimes>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> monom P \<one> m)"
ballarin@13940
   757
    by (simp add: UP_smult_assoc2)
ballarin@15095
   758
  also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"
ballarin@13940
   759
    by (simp add: monom_mult_smult del: r_one)
ballarin@15095
   760
  also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp
ballarin@13940
   761
  finally show ?thesis .
ballarin@13940
   762
qed
ballarin@13940
   763
ballarin@13940
   764
lemma (in UP_cring) monom_a_inv [simp]:
ballarin@15095
   765
  "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
ballarin@13940
   766
  by (rule up_eqI) simp_all
ballarin@13940
   767
ballarin@13940
   768
lemma (in UP_cring) monom_inj:
ballarin@13940
   769
  "inj_on (%a. monom P a n) (carrier R)"
ballarin@13940
   770
proof (rule inj_onI)
ballarin@13940
   771
  fix x y
ballarin@13940
   772
  assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
ballarin@13940
   773
  then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
ballarin@13940
   774
  with R show "x = y" by simp
ballarin@13940
   775
qed
ballarin@13940
   776
ballarin@13949
   777
subsection {* The degree function *}
ballarin@13940
   778
wenzelm@14651
   779
constdefs (structure R)
ballarin@15095
   780
  deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
wenzelm@14651
   781
  "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
ballarin@13940
   782
ballarin@13940
   783
lemma (in UP_cring) deg_aboveI:
wenzelm@14666
   784
  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
ballarin@13940
   785
  by (unfold deg_def P_def) (fast intro: Least_le)
ballarin@15095
   786
ballarin@13940
   787
(*
ballarin@13940
   788
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
ballarin@13940
   789
proof -
ballarin@13940
   790
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   791
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   792
  then show ?thesis ..
ballarin@13940
   793
qed
wenzelm@14666
   794
ballarin@13940
   795
lemma bound_coeff_obtain:
ballarin@13940
   796
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
ballarin@13940
   797
proof -
ballarin@13940
   798
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   799
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   800
  with prem show P .
ballarin@13940
   801
qed
ballarin@13940
   802
*)
ballarin@15095
   803
ballarin@13940
   804
lemma (in UP_cring) deg_aboveD:
ballarin@13940
   805
  "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
ballarin@13940
   806
proof -
ballarin@13940
   807
  assume R: "p \<in> carrier P" and "deg R p < m"
wenzelm@14666
   808
  from R obtain n where "bound \<zero> n (coeff P p)"
ballarin@13940
   809
    by (auto simp add: UP_def P_def)
ballarin@13940
   810
  then have "bound \<zero> (deg R p) (coeff P p)"
ballarin@13940
   811
    by (auto simp: deg_def P_def dest: LeastI)
wenzelm@14666
   812
  then show ?thesis ..
ballarin@13940
   813
qed
ballarin@13940
   814
ballarin@13940
   815
lemma (in UP_cring) deg_belowI:
ballarin@13940
   816
  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
ballarin@13940
   817
    and R: "p \<in> carrier P"
ballarin@13940
   818
  shows "n <= deg R p"
wenzelm@14666
   819
-- {* Logically, this is a slightly stronger version of
ballarin@15095
   820
   @{thm [source] deg_aboveD} *}
ballarin@13940
   821
proof (cases "n=0")
ballarin@13940
   822
  case True then show ?thesis by simp
ballarin@13940
   823
next
ballarin@13940
   824
  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
ballarin@13940
   825
  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
ballarin@13940
   826
  then show ?thesis by arith
ballarin@13940
   827
qed
ballarin@13940
   828
ballarin@13940
   829
lemma (in UP_cring) lcoeff_nonzero_deg:
ballarin@13940
   830
  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
ballarin@13940
   831
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   832
proof -
ballarin@13940
   833
  from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
ballarin@13940
   834
  proof -
ballarin@13940
   835
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
ballarin@13940
   836
      by arith
ballarin@15095
   837
(* TODO: why does simplification below not work with "1" *)
ballarin@13940
   838
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
ballarin@13940
   839
      by (unfold deg_def P_def) arith
ballarin@13940
   840
    then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
ballarin@13940
   841
    then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
ballarin@13940
   842
      by (unfold bound_def) fast
ballarin@13940
   843
    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
wenzelm@14666
   844
    then show ?thesis by auto
ballarin@13940
   845
  qed
ballarin@13940
   846
  with deg_belowI R have "deg R p = m" by fastsimp
ballarin@13940
   847
  with m_coeff show ?thesis by simp
ballarin@13940
   848
qed
ballarin@13940
   849
ballarin@13940
   850
lemma (in UP_cring) lcoeff_nonzero_nonzero:
ballarin@15095
   851
  assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   852
  shows "coeff P p 0 ~= \<zero>"
ballarin@13940
   853
proof -
ballarin@13940
   854
  have "EX m. coeff P p m ~= \<zero>"
ballarin@13940
   855
  proof (rule classical)
ballarin@13940
   856
    assume "~ ?thesis"
ballarin@15095
   857
    with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
ballarin@13940
   858
    with nonzero show ?thesis by contradiction
ballarin@13940
   859
  qed
ballarin@13940
   860
  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
ballarin@13940
   861
  then have "m <= deg R p" by (rule deg_belowI)
ballarin@13940
   862
  then have "m = 0" by (simp add: deg)
ballarin@13940
   863
  with coeff show ?thesis by simp
ballarin@13940
   864
qed
ballarin@13940
   865
ballarin@13940
   866
lemma (in UP_cring) lcoeff_nonzero:
ballarin@15095
   867
  assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   868
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   869
proof (cases "deg R p = 0")
ballarin@13940
   870
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
ballarin@13940
   871
next
ballarin@13940
   872
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
ballarin@13940
   873
qed
ballarin@13940
   874
ballarin@13940
   875
lemma (in UP_cring) deg_eqI:
ballarin@13940
   876
  "[| !!m. n < m ==> coeff P p m = \<zero>;
ballarin@13940
   877
      !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
ballarin@13940
   878
by (fast intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   879
ballarin@13940
   880
(* Degree and polynomial operations *)
ballarin@13940
   881
ballarin@13940
   882
lemma (in UP_cring) deg_add [simp]:
ballarin@13940
   883
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   884
  shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
ballarin@13940
   885
proof (cases "deg R p <= deg R q")
ballarin@13940
   886
  case True show ?thesis
wenzelm@14666
   887
    by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
ballarin@13940
   888
next
ballarin@13940
   889
  case False show ?thesis
ballarin@13940
   890
    by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
ballarin@13940
   891
qed
ballarin@13940
   892
ballarin@13940
   893
lemma (in UP_cring) deg_monom_le:
ballarin@13940
   894
  "a \<in> carrier R ==> deg R (monom P a n) <= n"
ballarin@13940
   895
  by (intro deg_aboveI) simp_all
ballarin@13940
   896
ballarin@13940
   897
lemma (in UP_cring) deg_monom [simp]:
ballarin@13940
   898
  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
ballarin@13940
   899
  by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   900
ballarin@13940
   901
lemma (in UP_cring) deg_const [simp]:
ballarin@13940
   902
  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
ballarin@13940
   903
proof (rule le_anti_sym)
ballarin@13940
   904
  show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
ballarin@13940
   905
next
ballarin@13940
   906
  show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
ballarin@13940
   907
qed
ballarin@13940
   908
ballarin@13940
   909
lemma (in UP_cring) deg_zero [simp]:
ballarin@15095
   910
  "deg R \<zero>\<^bsub>P\<^esub> = 0"
ballarin@13940
   911
proof (rule le_anti_sym)
ballarin@15095
   912
  show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   913
next
ballarin@15095
   914
  show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   915
qed
ballarin@13940
   916
ballarin@13940
   917
lemma (in UP_cring) deg_one [simp]:
ballarin@15095
   918
  "deg R \<one>\<^bsub>P\<^esub> = 0"
ballarin@13940
   919
proof (rule le_anti_sym)
ballarin@15095
   920
  show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   921
next
ballarin@15095
   922
  show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   923
qed
ballarin@13940
   924
ballarin@13940
   925
lemma (in UP_cring) deg_uminus [simp]:
ballarin@15095
   926
  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
ballarin@13940
   927
proof (rule le_anti_sym)
ballarin@15095
   928
  show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
ballarin@13940
   929
next
ballarin@15095
   930
  show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
ballarin@13940
   931
    by (simp add: deg_belowI lcoeff_nonzero_deg
ballarin@13940
   932
      inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
ballarin@13940
   933
qed
ballarin@13940
   934
ballarin@13940
   935
lemma (in UP_domain) deg_smult_ring:
ballarin@13940
   936
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   937
  deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   938
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
ballarin@13940
   939
ballarin@13940
   940
lemma (in UP_domain) deg_smult [simp]:
ballarin@13940
   941
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   942
  shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   943
proof (rule le_anti_sym)
ballarin@15095
   944
  show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   945
    by (rule deg_smult_ring)
ballarin@13940
   946
next
ballarin@15095
   947
  show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   948
  proof (cases "a = \<zero>")
ballarin@13940
   949
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
ballarin@13940
   950
qed
ballarin@13940
   951
ballarin@13940
   952
lemma (in UP_cring) deg_mult_cring:
ballarin@13940
   953
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   954
  shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
ballarin@13940
   955
proof (rule deg_aboveI)
ballarin@13940
   956
  fix m
ballarin@13940
   957
  assume boundm: "deg R p + deg R q < m"
ballarin@13940
   958
  {
ballarin@13940
   959
    fix k i
ballarin@13940
   960
    assume boundk: "deg R p + deg R q < k"
ballarin@13940
   961
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
ballarin@13940
   962
    proof (cases "deg R p < i")
ballarin@13940
   963
      case True then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   964
    next
ballarin@13940
   965
      case False with boundk have "deg R q < k - i" by arith
ballarin@13940
   966
      then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   967
    qed
ballarin@13940
   968
  }
ballarin@15095
   969
  with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
ballarin@13940
   970
qed (simp add: R)
ballarin@13940
   971
ballarin@13940
   972
lemma (in UP_domain) deg_mult [simp]:
ballarin@15095
   973
  "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   974
  deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
ballarin@13940
   975
proof (rule le_anti_sym)
ballarin@13940
   976
  assume "p \<in> carrier P" " q \<in> carrier P"
ballarin@15095
   977
  show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)
ballarin@13940
   978
next
ballarin@13940
   979
  let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
ballarin@15095
   980
  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   981
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
ballarin@15095
   982
  show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   983
  proof (rule deg_belowI, simp add: R)
ballarin@15095
   984
    have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@15095
   985
      = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@13940
   986
      by (simp only: ivl_disj_un_one)
ballarin@15095
   987
    also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@13940
   988
      by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
   989
        deg_aboveD less_add_diff R Pi_def)
ballarin@15095
   990
    also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
ballarin@13940
   991
      by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   992
    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
ballarin@13940
   993
      by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
   994
        ivl_disj_int_singleton deg_aboveD R Pi_def)
ballarin@15095
   995
    finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@13940
   996
      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
ballarin@15095
   997
    with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
ballarin@13940
   998
      by (simp add: integral_iff lcoeff_nonzero R)
ballarin@13940
   999
    qed (simp add: R)
ballarin@13940
  1000
  qed
ballarin@13940
  1001
ballarin@13940
  1002
lemma (in UP_cring) coeff_finsum:
ballarin@13940
  1003
  assumes fin: "finite A"
ballarin@13940
  1004
  shows "p \<in> A -> carrier P ==>
ballarin@15095
  1005
    coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
ballarin@13940
  1006
  using fin by induct (auto simp: Pi_def)
ballarin@13940
  1007
ballarin@13940
  1008
lemma (in UP_cring) up_repr:
ballarin@13940
  1009
  assumes R: "p \<in> carrier P"
ballarin@15095
  1010
  shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
ballarin@13940
  1011
proof (rule up_eqI)
ballarin@13940
  1012
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
  1013
  fix k
ballarin@13940
  1014
  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
ballarin@13940
  1015
    by simp
ballarin@15095
  1016
  show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
ballarin@13940
  1017
  proof (cases "k <= deg R p")
ballarin@13940
  1018
    case True
ballarin@15095
  1019
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
  1020
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
ballarin@13940
  1021
      by (simp only: ivl_disj_un_one)
ballarin@13940
  1022
    also from True
ballarin@15095
  1023
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
ballarin@13940
  1024
      by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
  1025
        ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
ballarin@13940
  1026
    also
ballarin@15095
  1027
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
ballarin@13940
  1028
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
  1029
    also have "... = coeff P p k"
ballarin@13940
  1030
      by (simp cong: finsum_cong add: setsum_Un_disjoint
wenzelm@14666
  1031
        ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
ballarin@13940
  1032
    finally show ?thesis .
ballarin@13940
  1033
  next
ballarin@13940
  1034
    case False
ballarin@15095
  1035
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
  1036
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
ballarin@13940
  1037
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
  1038
    also from False have "... = coeff P p k"
ballarin@13940
  1039
      by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
ballarin@13940
  1040
        coeff_finsum deg_aboveD R Pi_def)
ballarin@13940
  1041
    finally show ?thesis .
ballarin@13940
  1042
  qed
ballarin@13940
  1043
qed (simp_all add: R Pi_def)
ballarin@13940
  1044
ballarin@13940
  1045
lemma (in UP_cring) up_repr_le:
ballarin@13940
  1046
  "[| deg R p <= n; p \<in> carrier P |] ==>
ballarin@15095
  1047
  (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
ballarin@13940
  1048
proof -
ballarin@13940
  1049
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
  1050
  assume R: "p \<in> carrier P" and "deg R p <= n"
ballarin@15095
  1051
  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
ballarin@13940
  1052
    by (simp only: ivl_disj_un_one)
ballarin@13940
  1053
  also have "... = finsum P ?s {..deg R p}"
ballarin@13940
  1054
    by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
  1055
      deg_aboveD R Pi_def)
ballarin@13940
  1056
  also have "... = p" by (rule up_repr)
ballarin@13940
  1057
  finally show ?thesis .
ballarin@13940
  1058
qed
ballarin@13940
  1059
ballarin@13949
  1060
subsection {* Polynomials over an integral domain form an integral domain *}
ballarin@13940
  1061
ballarin@13940
  1062
lemma domainI:
ballarin@13940
  1063
  assumes cring: "cring R"
ballarin@13940
  1064
    and one_not_zero: "one R ~= zero R"
ballarin@13940
  1065
    and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
ballarin@13940
  1066
      b \<in> carrier R |] ==> a = zero R | b = zero R"
ballarin@13940
  1067
  shows "domain R"
ballarin@13940
  1068
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
ballarin@13940
  1069
    del: disjCI)
ballarin@13940
  1070
ballarin@13940
  1071
lemma (in UP_domain) UP_one_not_zero:
ballarin@15095
  1072
  "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1073
proof
ballarin@15095
  1074
  assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
ballarin@15095
  1075
  hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
ballarin@13940
  1076
  hence "\<one> = \<zero>" by simp
ballarin@13940
  1077
  with one_not_zero show "False" by contradiction
ballarin@13940
  1078
qed
ballarin@13940
  1079
ballarin@13940
  1080
lemma (in UP_domain) UP_integral:
ballarin@15095
  1081
  "[| p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1082
proof -
ballarin@13940
  1083
  fix p q
ballarin@15095
  1084
  assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1085
  show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1086
  proof (rule classical)
ballarin@15095
  1087
    assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
ballarin@15095
  1088
    with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
ballarin@13940
  1089
    also from pq have "... = 0" by simp
ballarin@13940
  1090
    finally have "deg R p + deg R q = 0" .
ballarin@13940
  1091
    then have f1: "deg R p = 0 & deg R q = 0" by simp
ballarin@15095
  1092
    from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
ballarin@13940
  1093
      by (simp only: up_repr_le)
ballarin@13940
  1094
    also from R have "... = monom P (coeff P p 0) 0" by simp
ballarin@13940
  1095
    finally have p: "p = monom P (coeff P p 0) 0" .
ballarin@15095
  1096
    from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
ballarin@13940
  1097
      by (simp only: up_repr_le)
ballarin@13940
  1098
    also from R have "... = monom P (coeff P q 0) 0" by simp
ballarin@13940
  1099
    finally have q: "q = monom P (coeff P q 0) 0" .
ballarin@15095
  1100
    from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
ballarin@13940
  1101
    also from pq have "... = \<zero>" by simp
ballarin@13940
  1102
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
ballarin@13940
  1103
    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
ballarin@13940
  1104
      by (simp add: R.integral_iff)
ballarin@15095
  1105
    with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
ballarin@13940
  1106
  qed
ballarin@13940
  1107
qed
ballarin@13940
  1108
ballarin@13940
  1109
theorem (in UP_domain) UP_domain:
ballarin@13940
  1110
  "domain P"
ballarin@13940
  1111
  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
ballarin@13940
  1112
ballarin@13940
  1113
text {*
ballarin@15095
  1114
  Instantiation of theorems from @{term domain}.
ballarin@13940
  1115
*}
ballarin@13940
  1116
ballarin@15095
  1117
(* TODO: this should be automated with an instantiation command. *)
ballarin@15095
  1118
ballarin@13940
  1119
lemmas (in UP_domain) UP_zero_not_one [simp] =
ballarin@13940
  1120
  domain.zero_not_one [OF UP_domain]
ballarin@13940
  1121
ballarin@13940
  1122
lemmas (in UP_domain) UP_integral_iff =
ballarin@13940
  1123
  domain.integral_iff [OF UP_domain]
ballarin@13940
  1124
ballarin@13940
  1125
lemmas (in UP_domain) UP_m_lcancel =
ballarin@13940
  1126
  domain.m_lcancel [OF UP_domain]
ballarin@13940
  1127
ballarin@13940
  1128
lemmas (in UP_domain) UP_m_rcancel =
ballarin@13940
  1129
  domain.m_rcancel [OF UP_domain]
ballarin@13940
  1130
ballarin@13940
  1131
lemma (in UP_domain) smult_integral:
ballarin@15095
  1132
  "[| a \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1133
  by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
ballarin@13940
  1134
    inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
ballarin@13940
  1135
wenzelm@14666
  1136
ballarin@13949
  1137
subsection {* Evaluation Homomorphism and Universal Property*}
ballarin@13940
  1138
wenzelm@14666
  1139
(* alternative congruence rule (possibly more efficient)
wenzelm@14666
  1140
lemma (in abelian_monoid) finsum_cong2:
wenzelm@14666
  1141
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
wenzelm@14666
  1142
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
wenzelm@14666
  1143
  sorry*)
wenzelm@14666
  1144
ballarin@13940
  1145
theorem (in cring) diagonal_sum:
ballarin@13940
  1146
  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
wenzelm@14666
  1147
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1148
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1149
proof -
ballarin@13940
  1150
  assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
ballarin@13940
  1151
  {
ballarin@13940
  1152
    fix j
ballarin@13940
  1153
    have "j <= n + m ==>
wenzelm@14666
  1154
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1155
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
ballarin@13940
  1156
    proof (induct j)
ballarin@13940
  1157
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
ballarin@13940
  1158
    next
wenzelm@14666
  1159
      case (Suc j)
ballarin@13940
  1160
      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
wenzelm@14666
  1161
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1162
      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
wenzelm@14666
  1163
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1164
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
wenzelm@14666
  1165
        using Suc by (auto intro!: funcset_mem [OF Rf])
ballarin@13940
  1166
      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
wenzelm@14666
  1167
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1168
      have R11: "g 0 \<in> carrier R"
wenzelm@14666
  1169
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1170
      from Suc show ?case
wenzelm@14666
  1171
        by (simp cong: finsum_cong add: Suc_diff_le a_ac
wenzelm@14666
  1172
          Pi_def R6 R8 R9 R10 R11)
ballarin@13940
  1173
    qed
ballarin@13940
  1174
  }
ballarin@13940
  1175
  then show ?thesis by fast
ballarin@13940
  1176
qed
ballarin@13940
  1177
ballarin@13940
  1178
lemma (in abelian_monoid) boundD_carrier:
ballarin@13940
  1179
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
ballarin@13940
  1180
  by auto
ballarin@13940
  1181
ballarin@13940
  1182
theorem (in cring) cauchy_product:
ballarin@13940
  1183
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
ballarin@13940
  1184
    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
wenzelm@14666
  1185
  shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
ballarin@15095
  1186
    (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"       (* State revese direction? *)
ballarin@13940
  1187
proof -
ballarin@13940
  1188
  have f: "!!x. f x \<in> carrier R"
ballarin@13940
  1189
  proof -
ballarin@13940
  1190
    fix x
ballarin@13940
  1191
    show "f x \<in> carrier R"
ballarin@13940
  1192
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
ballarin@13940
  1193
  qed
ballarin@13940
  1194
  have g: "!!x. g x \<in> carrier R"
ballarin@13940
  1195
  proof -
ballarin@13940
  1196
    fix x
ballarin@13940
  1197
    show "g x \<in> carrier R"
ballarin@13940
  1198
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
ballarin@13940
  1199
  qed
wenzelm@14666
  1200
  from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1201
      (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1202
    by (simp add: diagonal_sum Pi_def)
nipkow@15045
  1203
  also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1204
    by (simp only: ivl_disj_un_one)
wenzelm@14666
  1205
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1206
    by (simp cong: finsum_cong
wenzelm@14666
  1207
      add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@15095
  1208
  also from f g
ballarin@15095
  1209
  have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1210
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
wenzelm@14666
  1211
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
ballarin@13940
  1212
    by (simp cong: finsum_cong
wenzelm@14666
  1213
      add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1214
  also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
ballarin@13940
  1215
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
ballarin@13940
  1216
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
ballarin@13940
  1217
  finally show ?thesis .
ballarin@13940
  1218
qed
ballarin@13940
  1219
ballarin@13940
  1220
lemma (in UP_cring) const_ring_hom:
ballarin@13940
  1221
  "(%a. monom P a 0) \<in> ring_hom R P"
ballarin@13940
  1222
  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
ballarin@13940
  1223
wenzelm@14651
  1224
constdefs (structure S)
ballarin@15095
  1225
  eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
ballarin@15095
  1226
           'a => 'b, 'b, nat => 'a] => 'b"
wenzelm@14651
  1227
  "eval R S phi s == \<lambda>p \<in> carrier (UP R).
ballarin@15095
  1228
    \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"
ballarin@15095
  1229
ballarin@15095
  1230
locale UP_univ_prop = ring_hom_cring R S + UP_cring R
wenzelm@14666
  1231
ballarin@15095
  1232
lemma (in UP) eval_on_carrier:
ballarin@15095
  1233
  includes struct S
ballarin@15095
  1234
  shows  "p \<in> carrier P ==>
ballarin@13940
  1235
    eval R S phi s p =
ballarin@15095
  1236
    (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1237
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1238
ballarin@15095
  1239
lemma (in UP) eval_extensional:
ballarin@13940
  1240
  "eval R S phi s \<in> extensional (carrier P)"
ballarin@13940
  1241
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1242
ballarin@15095
  1243
theorem (in UP_univ_prop) eval_ring_hom:
ballarin@13940
  1244
  "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
ballarin@13940
  1245
proof (rule ring_hom_memI)
ballarin@13940
  1246
  fix p
ballarin@13940
  1247
  assume RS: "p \<in> carrier P" "s \<in> carrier S"
ballarin@13940
  1248
  then show "eval R S h s p \<in> carrier S"
ballarin@13940
  1249
    by (simp only: eval_on_carrier) (simp add: Pi_def)
ballarin@13940
  1250
next
ballarin@13940
  1251
  fix p q
ballarin@13940
  1252
  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
ballarin@15095
  1253
  then show "eval R S h s (p \<otimes>\<^bsub>P\<^esub> q) = eval R S h s p \<otimes>\<^bsub>S\<^esub> eval R S h s q"
ballarin@13940
  1254
  proof (simp only: eval_on_carrier UP_mult_closed)
ballarin@13940
  1255
    from RS have
ballarin@15095
  1256
      "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
ballarin@15095
  1257
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<otimes>\<^bsub>P\<^esub> q)<..deg R p + deg R q}.
ballarin@15095
  1258
        h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1259
      by (simp cong: finsum_cong
wenzelm@14666
  1260
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
wenzelm@14666
  1261
        del: coeff_mult)
ballarin@13940
  1262
    also from RS have "... =
ballarin@15095
  1263
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1264
      by (simp only: ivl_disj_un_one deg_mult_cring)
ballarin@13940
  1265
    also from RS have "... =
ballarin@15095
  1266
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
ballarin@15095
  1267
         \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
ballarin@15095
  1268
           h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
ballarin@15095
  1269
           (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
ballarin@13940
  1270
      by (simp cong: finsum_cong add: nat_pow_mult Pi_def
wenzelm@14666
  1271
        S.m_ac S.finsum_rdistr)
ballarin@13940
  1272
    also from RS have "... =
ballarin@15095
  1273
      (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
ballarin@15095
  1274
      (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
wenzelm@14666
  1275
      by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
wenzelm@14666
  1276
        Pi_def)
ballarin@13940
  1277
    finally show
ballarin@15095
  1278
      "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
ballarin@15095
  1279
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
ballarin@15095
  1280
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
ballarin@13940
  1281
  qed
ballarin@13940
  1282
next
ballarin@13940
  1283
  fix p q
ballarin@13940
  1284
  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
ballarin@15095
  1285
  then show "eval R S h s (p \<oplus>\<^bsub>P\<^esub> q) = eval R S h s p \<oplus>\<^bsub>S\<^esub> eval R S h s q"
ballarin@13940
  1286
  proof (simp only: eval_on_carrier UP_a_closed)
ballarin@13940
  1287
    from RS have
ballarin@15095
  1288
      "(\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
ballarin@15095
  1289
      (\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<oplus>\<^bsub>P\<^esub> q)<..max (deg R p) (deg R q)}.
ballarin@15095
  1290
        h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1291
      by (simp cong: finsum_cong
wenzelm@14666
  1292
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
wenzelm@14666
  1293
        del: coeff_add)
ballarin@13940
  1294
    also from RS have "... =
ballarin@15095
  1295
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
ballarin@15095
  1296
          h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1297
      by (simp add: ivl_disj_un_one)
ballarin@13940
  1298
    also from RS have "... =
ballarin@15095
  1299
      (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1300
      (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1301
      by (simp cong: finsum_cong
wenzelm@14666
  1302
        add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1303
    also have "... =
ballarin@15095
  1304
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
ballarin@15095
  1305
          h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1306
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
ballarin@15095
  1307
          h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1308
      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
ballarin@13940
  1309
    also from RS have "... =
ballarin@15095
  1310
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1311
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1312
      by (simp cong: finsum_cong
wenzelm@14666
  1313
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1314
    finally show
ballarin@15095
  1315
      "(\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
ballarin@15095
  1316
      (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1317
      (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
ballarin@13940
  1318
  qed
ballarin@13940
  1319
next
ballarin@13940
  1320
  assume S: "s \<in> carrier S"
ballarin@15095
  1321
  then show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
ballarin@13940
  1322
    by (simp only: eval_on_carrier UP_one_closed) simp
ballarin@13940
  1323
qed
ballarin@13940
  1324
ballarin@13940
  1325
text {* Instantiation of ring homomorphism lemmas. *}
ballarin@13940
  1326
ballarin@15095
  1327
(* TODO: again, automate with instantiation command *)
ballarin@15095
  1328
ballarin@15095
  1329
lemma (in UP_univ_prop) ring_hom_cring_P_S:
ballarin@13940
  1330
  "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
ballarin@13940
  1331
  by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
ballarin@15095
  1332
    intro: ring_hom_cring_axioms.intro eval_ring_hom)
ballarin@13940
  1333
ballarin@15095
  1334
(*
ballarin@15095
  1335
lemma (in UP_univ_prop) UP_hom_closed [intro, simp]:
ballarin@13940
  1336
  "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
ballarin@13940
  1337
  by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
ballarin@13940
  1338
ballarin@15095
  1339
lemma (in UP_univ_prop) UP_hom_mult [simp]:
ballarin@13940
  1340
  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
  1341
  eval R S h s (p \<otimes>\<^bsub>P\<^esub> q) = eval R S h s p \<otimes>\<^bsub>S\<^esub> eval R S h s q"
ballarin@13940
  1342
  by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
ballarin@13940
  1343
ballarin@15095
  1344
lemma (in UP_univ_prop) UP_hom_add [simp]:
ballarin@13940
  1345
  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
  1346
  eval R S h s (p \<oplus>\<^bsub>P\<^esub> q) = eval R S h s p \<oplus>\<^bsub>S\<^esub> eval R S h s q"
ballarin@13940
  1347
  by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
ballarin@13940
  1348
ballarin@15095
  1349
lemma (in UP_univ_prop) UP_hom_one [simp]:
ballarin@15095
  1350
  "s \<in> carrier S ==> eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
ballarin@13940
  1351
  by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
ballarin@13940
  1352
ballarin@15095
  1353
lemma (in UP_univ_prop) UP_hom_zero [simp]:
ballarin@15095
  1354
  "s \<in> carrier S ==> eval R S h s \<zero>\<^bsub>P\<^esub> = \<zero>\<^bsub>S\<^esub>"
ballarin@13940
  1355
  by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
ballarin@13940
  1356
ballarin@15095
  1357
lemma (in UP_univ_prop) UP_hom_a_inv [simp]:
ballarin@13940
  1358
  "[| s \<in> carrier S; p \<in> carrier P |] ==>
ballarin@15095
  1359
  (eval R S h s) (\<ominus>\<^bsub>P\<^esub> p) = \<ominus>\<^bsub>S\<^esub> (eval R S h s) p"
ballarin@13940
  1360
  by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
ballarin@13940
  1361
ballarin@15095
  1362
lemma (in UP_univ_prop) UP_hom_finsum [simp]:
ballarin@13940
  1363
  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
ballarin@13940
  1364
  (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
ballarin@13940
  1365
  by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
ballarin@13940
  1366
ballarin@15095
  1367
lemma (in UP_univ_prop) UP_hom_finprod [simp]:
ballarin@13940
  1368
  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
ballarin@13940
  1369
  (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
ballarin@13940
  1370
  by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
ballarin@15095
  1371
*)
ballarin@13940
  1372
ballarin@13940
  1373
text {* Further properties of the evaluation homomorphism. *}
ballarin@13940
  1374
ballarin@13940
  1375
(* The following lemma could be proved in UP\_cring with the additional
ballarin@13940
  1376
   assumption that h is closed. *)
ballarin@13940
  1377
ballarin@15095
  1378
lemma (in UP_univ_prop) eval_const:
ballarin@13940
  1379
  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
ballarin@13940
  1380
  by (simp only: eval_on_carrier monom_closed) simp
ballarin@13940
  1381
ballarin@13940
  1382
text {* The following proof is complicated by the fact that in arbitrary
ballarin@13940
  1383
  rings one might have @{term "one R = zero R"}. *}
ballarin@13940
  1384
ballarin@13940
  1385
(* TODO: simplify by cases "one R = zero R" *)
ballarin@13940
  1386
ballarin@15095
  1387
lemma (in UP_univ_prop) eval_monom1:
ballarin@13940
  1388
  "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
ballarin@13940
  1389
proof (simp only: eval_on_carrier monom_closed R.one_closed)
ballarin@13940
  1390
  assume S: "s \<in> carrier S"
wenzelm@14666
  1391
  then have
ballarin@15095
  1392
    "(\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
ballarin@15095
  1393
    (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
ballarin@15095
  1394
      h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1395
    by (simp cong: finsum_cong del: coeff_monom
ballarin@13940
  1396
      add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1397
  also have "... =
ballarin@15095
  1398
    (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1399
    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
ballarin@13940
  1400
  also have "... = s"
ballarin@15095
  1401
  proof (cases "s = \<zero>\<^bsub>S\<^esub>")
ballarin@13940
  1402
    case True then show ?thesis by (simp add: Pi_def)
ballarin@13940
  1403
  next
ballarin@13940
  1404
    case False with S show ?thesis by (simp add: Pi_def)
ballarin@13940
  1405
  qed
ballarin@15095
  1406
  finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
ballarin@15095
  1407
    h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
ballarin@13940
  1408
qed
ballarin@13940
  1409
ballarin@13940
  1410
lemma (in UP_cring) monom_pow:
ballarin@13940
  1411
  assumes R: "a \<in> carrier R"
ballarin@15095
  1412
  shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
ballarin@13940
  1413
proof (induct m)
ballarin@13940
  1414
  case 0 from R show ?case by simp
ballarin@13940
  1415
next
ballarin@13940
  1416
  case Suc with R show ?case
ballarin@13940
  1417
    by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
ballarin@13940
  1418
qed
ballarin@13940
  1419
ballarin@13940
  1420
lemma (in ring_hom_cring) hom_pow [simp]:
ballarin@15095
  1421
  "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
ballarin@13940
  1422
  by (induct n) simp_all
ballarin@13940
  1423
ballarin@15095
  1424
lemma (in UP_univ_prop) eval_monom:
ballarin@13940
  1425
  "[| s \<in> carrier S; r \<in> carrier R |] ==>
ballarin@15095
  1426
  eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@13940
  1427
proof -
ballarin@15095
  1428
  assume S: "s \<in> carrier S" and R: "r \<in> carrier R"
ballarin@15696
  1429
  from ring_hom_cring_P_S [OF S] interpret ring_hom_cring [P S "eval R S h s"]
ballarin@15696
  1430
    by - (rule ring_hom_cring.axioms, assumption)+
ballarin@15696
  1431
    (* why is simplifier invoked --- in done ??? *)
ballarin@15095
  1432
  from R S have "eval R S h s (monom P r n) =
ballarin@15095
  1433
    eval R S h s (monom P r 0 \<otimes>\<^bsub>P\<^esub> (monom P \<one> 1) (^)\<^bsub>P\<^esub> n)"
ballarin@15095
  1434
    by (simp del: monom_mult (* eval.hom_mult eval.hom_pow, delayed inst! *)
ballarin@13940
  1435
      add: monom_mult [THEN sym] monom_pow)
ballarin@15095
  1436
  also
ballarin@15095
  1437
  from R S eval_monom1 have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@13940
  1438
    by (simp add: eval_const)
ballarin@13940
  1439
  finally show ?thesis .
ballarin@13940
  1440
qed
ballarin@13940
  1441
ballarin@15095
  1442
lemma (in UP_univ_prop) eval_smult:
ballarin@13940
  1443
  "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
  1444
  eval R S h s (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> eval R S h s p"
ballarin@15095
  1445
proof -
ballarin@15095
  1446
  assume S: "s \<in> carrier S" and R: "r \<in> carrier R" and P: "p \<in> carrier P"
ballarin@15696
  1447
  from ring_hom_cring_P_S [OF S] interpret ring_hom_cring [P S "eval R S h s"]
ballarin@15696
  1448
    by - (rule ring_hom_cring.axioms, assumption)+
ballarin@15095
  1449
  from S R P show ?thesis
ballarin@15095
  1450
    by (simp add: monom_mult_is_smult [THEN sym] eval_const)
ballarin@15095
  1451
qed
ballarin@13940
  1452
ballarin@13940
  1453
lemma ring_hom_cringI:
ballarin@13940
  1454
  assumes "cring R"
ballarin@13940
  1455
    and "cring S"
ballarin@13940
  1456
    and "h \<in> ring_hom R S"
ballarin@13940
  1457
  shows "ring_hom_cring R S h"
ballarin@13940
  1458
  by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
ballarin@13940
  1459
    cring.axioms prems)
ballarin@13940
  1460
ballarin@15095
  1461
lemma (in UP_univ_prop) UP_hom_unique:
ballarin@13940
  1462
  assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1463
      "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
ballarin@13940
  1464
    and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1465
      "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
ballarin@15095
  1466
    and S: "s \<in> carrier S" and P: "p \<in> carrier P"
ballarin@13940
  1467
  shows "Phi p = Psi p"
ballarin@13940
  1468
proof -
ballarin@15696
  1469
  from UP_cring interpret cring [P] by - (rule cring.axioms, assumption)+
ballarin@15696
  1470
  interpret Phi: ring_hom_cring [P S Phi]
ballarin@15696
  1471
    by (auto intro: ring_hom_cring.axioms ring_hom_cringI UP_cring S.cring Phi)
ballarin@15696
  1472
  interpret Psi: ring_hom_cring [P S Psi]
ballarin@15696
  1473
    by (auto intro: ring_hom_cring.axioms ring_hom_cringI UP_cring S.cring Psi)
ballarin@15696
  1474
ballarin@15095
  1475
  have "Phi p =
ballarin@15095
  1476
      Phi (\<Oplus>\<^bsub>P \<^esub>i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
ballarin@15095
  1477
    by (simp add: up_repr P S monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@15696
  1478
  also
ballarin@15696
  1479
  have "... =
ballarin@15095
  1480
      Psi (\<Oplus>\<^bsub>P \<^esub>i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
ballarin@15095
  1481
    by (simp add: Phi Psi P S Pi_def comp_def)
ballarin@15696
  1482
(* Without interpret, the following command would have been necessary.
wenzelm@14666
  1483
    by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
ballarin@13940
  1484
      ring_hom_cring.hom_mult [OF Phi_hom]
ballarin@13940
  1485
      ring_hom_cring.hom_pow [OF Phi_hom] Phi
wenzelm@14666
  1486
      ring_hom_cring.hom_finsum [OF Psi_hom]
ballarin@13940
  1487
      ring_hom_cring.hom_mult [OF Psi_hom]
ballarin@13940
  1488
      ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
ballarin@15095
  1489
*)
ballarin@13940
  1490
  also have "... = Psi p"
ballarin@15095
  1491
    by (simp add: up_repr P S monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@13940
  1492
  finally show ?thesis .
ballarin@13940
  1493
qed
ballarin@13940
  1494
ballarin@15095
  1495
theorem (in UP_univ_prop) UP_universal_property:
ballarin@13940
  1496
  "s \<in> carrier S ==>
ballarin@13940
  1497
  EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
wenzelm@14666
  1498
    Phi (monom P \<one> 1) = s &
ballarin@13940
  1499
    (ALL r : carrier R. Phi (monom P r 0) = h r)"
wenzelm@14666
  1500
  using eval_monom1
ballarin@13940
  1501
  apply (auto intro: eval_ring_hom eval_const eval_extensional)
wenzelm@14666
  1502
  apply (rule extensionalityI)
wenzelm@14666
  1503
  apply (auto intro: UP_hom_unique)
wenzelm@14666
  1504
  done
ballarin@13940
  1505
ballarin@13940
  1506
subsection {* Sample application of evaluation homomorphism *}
ballarin@13940
  1507
ballarin@15095
  1508
lemma UP_univ_propI:
ballarin@13940
  1509
  assumes "cring R"
ballarin@13940
  1510
    and "cring S"
ballarin@13940
  1511
    and "h \<in> ring_hom R S"
ballarin@15095
  1512
  shows "UP_univ_prop R S h"
ballarin@15095
  1513
  by (fast intro: UP_univ_prop.intro ring_hom_cring_axioms.intro
ballarin@13940
  1514
    cring.axioms prems)
ballarin@13940
  1515
ballarin@13975
  1516
constdefs
ballarin@13975
  1517
  INTEG :: "int ring"
ballarin@13975
  1518
  "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
ballarin@13975
  1519
ballarin@15095
  1520
lemma INTEG_cring:
ballarin@13975
  1521
  "cring INTEG"
ballarin@13975
  1522
  by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
ballarin@13975
  1523
    zadd_zminus_inverse2 zadd_zmult_distrib)
ballarin@13975
  1524
ballarin@15095
  1525
lemma INTEG_id_eval:
ballarin@15095
  1526
  "UP_univ_prop INTEG INTEG id"
ballarin@15095
  1527
  by (fast intro: UP_univ_propI INTEG_cring id_ring_hom)
ballarin@13940
  1528
ballarin@13940
  1529
text {*
ballarin@15763
  1530
  Interpretation allows now to import all theorems and lemmas
ballarin@13940
  1531
  valid in the context of homomorphisms between @{term INTEG} and @{term
ballarin@15095
  1532
  "UP INTEG"} globally.
wenzelm@14666
  1533
*}
ballarin@13940
  1534
ballarin@15763
  1535
interpretation INTEG: UP_univ_prop [INTEG INTEG id]
ballarin@15763
  1536
  using INTEG_id_eval
ballarin@15763
  1537
  by - (rule UP_univ_prop.axioms, assumption)+
ballarin@15763
  1538
ballarin@13940
  1539
lemma INTEG_closed [intro, simp]:
ballarin@13940
  1540
  "z \<in> carrier INTEG"
ballarin@13940
  1541
  by (unfold INTEG_def) simp
ballarin@13940
  1542
ballarin@13940
  1543
lemma INTEG_mult [simp]:
ballarin@13940
  1544
  "mult INTEG z w = z * w"
ballarin@13940
  1545
  by (unfold INTEG_def) simp
ballarin@13940
  1546
ballarin@13940
  1547
lemma INTEG_pow [simp]:
ballarin@13940
  1548
  "pow INTEG z n = z ^ n"
ballarin@13940
  1549
  by (induct n) (simp_all add: INTEG_def nat_pow_def)
ballarin@13940
  1550
ballarin@13940
  1551
lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
ballarin@15763
  1552
  by (simp add: INTEG.eval_monom)
ballarin@13940
  1553
wenzelm@14590
  1554
end